This invention relates generally to plasma devices and particularly to the confinement and stabilization of plasmas in fusion devices by means of average magnetic well. More particularly, the present invention relates to the combination of plasma cross section shaping and the plasma pinch effect at small aspect ratio for the production of average magnetic well in a toroidal reversed field pinch.
Toroidal plasma devices are devices in which plasma is created in a topologically toroidal space, usually axisymmetric, and is confined therein by appropriate confining magnetic fields. Toroidal plasma devices are useful in the generation, confinement, heating, study and analysis of plasmas. In particular, such devices are useful for the reaction of deuterium and tritium, deuterium and deuterium or other nuclear fusible mixtures, with the production of high energy neutrons and energetic charged particles as products of the nuclear fusion reactions.
The problems in nuclear fusion devices are largely to heat the plasma to a temperature high enough to enable the desired reactions to occur and to confine the heated plasma for a time long enough to release energy in excess of that required to heat the plasma to reaction temperature. The present invention is directed to the magnetic confinement of such plasma and finds particular utility in such devices and their applications, including experimental devices and the use thereof in experimentation and investigation with respect to toroidal plasma devices.
A number of toroidal plasma devices have been suggested and built. The ones most closely related to the present invention are tokamak devices and pinch devices, including reversed field pinch (RFP) devices. In such devices, gas is confined in a toroidal confinement vessel and is heated to form a plasma which is generally held away from the walls of the confinement vessel by appropriate magnetic fields. Such devices are all topologically toroidal and are usually axisymmetric. A topological torus is any geometric solid figure that can be produced by an imagined elastic deformation of an initial circular torus. An axisymmetric torus is obtained by rotating any plane geometric figure about the major toroidal axis. An axisymmetric toroidal device is one in which all quantities are invariant to rotation about the major toroidal axis. A necessary condition for the toroidal magnetic confinement of plasmas is that the complete set of magnetic field components results in a set of nested, toroidally closed magnetic surfaces. A magnetic surface is defined as a mathematical surface on which the magnetic field has no component normal thereto. The magnetic surface enclosing zero volume in the center of the nest is called an elliptic magnetic axis. Most devices have only a single elliptic magnetic axis and a single set of nested surfaces. However, Doublet devices have two elliptic magnetic axes, and multipole devices have two or more sets of nested surfaces.
In some toroidal devices, such as tokamak and pinch devices, the confining magnetic field includes magnetic field components produced by currents flowing through the confined plasma itself. When nested magnetic surfaces are present, this current is significantly concentrated into those magnetic surfaces closer to elliptic magnetic axes. Such regions of greater current density relative to the remainder of the plasma are called current channels.
In those toroidal devices where it is required, a toroidal plasma current is usually produced by a transformer with the toroidal confined plasma acting as the secondary and with the primary being a central solenoid. Upon change of the current in the solenoid, a toroidal electric field is produced to ionize the gas and drive plasma current around the torus.
A pinch effect takes place when electric current flowing through the plasma is acted upon by its own magnetic field to exert a confining pressure on the plasma. The large current simultaneously heats the plasma ohmically. However, this simplest configuration by itself, called the Bennett pinch, is unstable, and most of the plasma soon strikes the confinement vessel, hence cooling the plasma and impeding any reaction. For this reason, additional measures are taken to improve the stability of the system.
The magnetohydrodynamic (MHD) stability of a magnetically confined plasma is dependent on the pitch of the magnetic field lines encircling the magnetic axis or axes. This pitch P is defined by ##EQU1## where .DELTA..zeta. is the distance traversed along the direction of the magnetic axis and k the number of times the axis is encircled, both while following a field line. This limit is the same for all possible field lines on a given magnetic surface. In toroidal plasma devices it is customary to use instead the safety factor q where ##EQU2## Here &lt;R&gt; is the average major radius of the magnetic surface in question. R is the major radius measured radially from the major toroidal axis to the magnetic surface. The aspect ratio A of a torus is defined by A.tbd.R.sub.o /a where R.sub.o is the major radius to the elliptic magnetic axis and a is the mean minor radius of the plasma surface. For a general topological torus &lt;R&gt;=&lt;C&gt;/2.pi., where &lt;C&gt; is the average major circumference of the nonaxisymmetric magnetic surface in question. There is a corresponding relationship between pitch P and safety factor q for still more general systems. In order to be magnetohydrodynamically stable, toroidal plasma devices must satisfy certain necessary conditions on q. If r is the mean minor radius, then these conditions are usually simply stated as: ##EQU3## must be large enough to satisfy relevant criteria, including the Mercier and the Robinson criteria; in particular, dq/dr must not change sign within the plasma, and it may be zero only at a magnetic axis. Conditions (a) and (b) taken together for large aspect ratio devices require that in plasmas with current channels, such as tokamaks and pinches, either .vertline.q.vertline..gtoreq.1 on axis and increases monotonically everywhere else in the plasma; or else .vertline.q.vertline.&lt;1 everywhere, decreases monotonically with increasing distance from the magnetic axis, passes smoothly through zero, and then increases monotonically with increasing distance from the magnetic axis in the outside regions of the plasma. The .vertline.q.vertline..gtoreq.1 case is realized in tokamaks, and the .vertline.q.vertline.&lt;1 case in reversed field pinches. Condition (a) above is usually required to avoid a serious kink instability that arises when q.apprxeq.1. A more general criterion for kink mode stability is given by the energy principle of I. B. Bernstein, et. al., in Proceedings of The Royal Society of London, A, 244, (1958), pp. 17-40. For low values of poloidal beta .beta..sub.p, defined by .beta..sub.p .tbd.2.mu..sub.o p/B.sub.P.sup.2, it is possible to find plasma equilibria which are stable to kink modes even when q=1. Here p and B.sub.P.sup.2 are the average over the plasma volume of the pressure p and the square of the poloidal magnetic field intensity B.sub.P, and .mu..sub.o is the vacuum magnetic permeability. For example, when A&lt;2/.beta..sub.p, condition (a) is not always required.
In the case of arbitrarily shaped flux surfaces in axisymmetric tori, Eq. (2) can be written in the easily applied form ##EQU4## where B.sub.T is the toroidal and B.sub.P the poloidal magnetic field intensity. The closed line integral, where l is the poloidal arc length, is taken around the flux surface at a constant toroidal angle. The convention used here is that B.sub.T &gt;0 on the elliptic magnetic axis of the reverse field pinch. Thus q&gt;0 on such axis and monotonically decreases with increasing distance from such axis, changing sign at the field reversal point. In the opposite convention, with B.sub.T &lt;0 on such axis, then q&lt;0 on such axis and monotonically increases with increasing distance from such axis. The quantity s appearing in condition (b) above is the magnetic shear, which exerts a stabilizing effect on many classes of instabilities, particularly on ideal MHD interchange instabilities and on many microinstabilities.
Another important property, which enhances stability by suppressing those MHD instabilities that are excited specifically by plasma pressure, is average magnetic well or minimum average B, where B is the magnetic field intensity. A review of the advantages of average magnetic well and of many configurations that have this property is given by H. P. Furth in Advances in Plasma Physics, Simon and Thompson, eds., 1 (Interscience Publishers, New York, 1968), pp. 67-100. The average square of the magnetic field intensity &lt;B.sup.2 &gt; on a flux surface is calculated by ##EQU5## where the integration is taken by following a magnetic field line for a sufficient distance to sample all of the magnetic surface. The simplest definition of average magnetic well in the limit where the plasma pressure is small is a minimum of &lt;B.sup.2 &gt; within the plasma. More generally, an average magnetic wall exists when there is a minimum in EQU &lt;B.sup.2 &gt;+2.mu..sub.o p. (5)
Condition (5) also has a strong correlation to the stability of resistive interchange modes in reverse field pinch configurations. When &lt;B.sup.2 &gt;2.mu..sub.o p increases with increasing distance from the elliptic magnetic axis, the resistive interchange mode is stable.
Average magnetic well implies that the average of the magnitude of the magnetic field increases outwardly from the center of the device. Therefore, if the plasma is driven outward by an incipient instability, it encounters a stronger magnetic field which opposes its outward motion.
The most commonly used toroidal magnetic confinement configuration at present is the tokamak, whose principal defining characteristic is to satisfy the q stability requirements by operating with .vertline.q.vertline.&gt;1 and s.gtoreq.0 by supplying a sufficiently large toroidal magnetic field intensity B.sub.T, in accordance with Eq. (3). Because the aspect ratio A is generally .gtoreq.3, the toroidal field, which must be provided by a large toroidal field coil system disposed around the confinement vessel, must be large. Typically, B.sub.T =5 B.sub.P to 10 B.sub.P. Therefore, the maximum toroidal current I.sub.p flowing in the plasma, which is related to poloidal magnetic field intensity B.sub.P by the formula B.sub.P =.mu..sub.o I.sub.p /2.pi.r, and with it the ohmic heating of the plasma, are limited by the maximum possible toroidal field intensity B.sub.T that can be withstood in a practical magnet system. A small magnetic well, which is also important for tokamak stability, is obtained by toroidal effects. The theoretically predicted maximum plasma pressure that can be confined is limited to .beta..ltorsim.0.10 and may well be less, where .beta..tbd.p(B.sup.2 /2.mu..sub.o) is the ratio of the volume averaged plasma pressure to the magnetic pressure of the confining field. (Here and throughout the remainder of this disclosure the SI mks system of units is used.) Because of the small .beta. of the tokamak, fusion reactor concepts based on it either must be large or must employ extraordinarily high toroidal magnetic field strength.
The toroidal magnetic field produced by the toroidal field coil system is referred to as a vacuum toroidal field when no plasma is present. The toroidal magnetic field then varies inversely with major radius and the quantity of f.tbd.RB.sub.T is a constant. When f is independent of which flux surface is under consideration in a region in the plasma, the toroidal magnetic field in that region is said to be a vacuum magnetic field. That is, the toroidal magnetic field in a plasma is similar to a vacuum toroidal magnetic field when B.sub.T varies inversely with R.
Pinches are most readily distinguished from tokamaks, which they superficially resemble, by having .vertline.q.vertline.&lt;1 everywhere throughout the plasma, and usually they have .vertline.q.vertline.&lt;&lt;1. A toroidal pinch previously known to satisfy the necessary conditions on q is the reversed field pinch (RFP). A recent review of the RFP art has been given by H. A. B. Bodin and A. A. Newton, Nucl. Fusion 20 (1980), pp. 1255-1324. The RFP is a diffuse pinch in which the magnetic field component sensibly parallel to the magnetic axis has a direction in the outside region of the plasma opposite to that in the inner region, and as a result, q(r) passes through zero and changes sign within the plasma. In fact, greatly reduced instability is observed in pinch experiments when the reversed q(r) profile is established. The field and q reversal is achieved by trapping a toroidal field in a pinched plasma and providing external boundary conditions such that a toroidal field of the opposite sign can exist between the plasma and the wall. A conducting shell is also required for stability. The combination of toroidal current and reversed toroidal magnetic field achieved in RFPs produces an equilibrium state of very low free energy, which is stable at low .beta.. This stability is independent of toroidal effects. Therefore, RFP aspect ratios can be chosen at will to optimize engineering and reactor parameters.
In the RFP the externally acting toroidal field is smaller than B.sub.P. Therefore, unlike in the tokamak, I.sub.p is limited only by the maximum intensity of B.sub.P that can be withstood in the device, and large ohmic heating of the plasma is possible. Furthermore, the maximum .beta. achievable in RFP devices will be greater than in tokamaks. Therefore, fusion reactor concepts based on the RFP can either be smaller or use lower magnetic fields than with tokamaks.
Unfortunately, the RFP does not possess a magnetic well, and it has been predicted theoretically and observed in computer plasma simulations that an m=0 resistive interchange instability grows into a large convective cell near the q=0 surface and limits plasma confinement. Here m is the poloidal mode number of the instability in question. There are data suggesting that this instability is present in contemporary RFP experiments. Resistive interchange instabilities are among those that can be stabilized by magnetic well.
Multipole plasma confinement devices take a different approach to toroidal plasma confinement. In multipole devices, the toroidal plasma current is replaced by two or more solid conducting rings located internal to the plasma, which produce a set of nested closed magnetic surfaces around each ring. By convention the number of poles is equal to twice the number of conductors. Thus, for example, a device with two internal conductors is termed a quadrupole; four an octopole, etc. Since the current flows through rigid conductors, the current flow is stable. There is no necessity for a strong toroidal magnetic field. The current rings are placed so as to generate a multipolar magnetic field and at least one hyperbolic magnetic axis within the space roughly enclosed poloidally by the rings. A hyperbolic magnetic axis occurs on a flux surface when there are more than two possible directions in which the magnetic field line may be traced. Furthermore, these rings and the hyperbolic magnetic axis or axes are surrounded by an outer set of nested closed magnetic surfaces. The magnetic surface or surfaces passing through the hyperbolic magnetic axis and separating the outer magnetic surfaces from those magnetic surfaces that enclose only a single ring are called separatrix magnetic surfaces. Excellent confinement has been demonstrated in experimental multipole devices. Shear can be added by means of only a small toroidal field.
Mulipole devices have a number of serious difficulties for high temperature plasma and fusion applications associated with the placement of conducting rings internal to the plasma. The rings require support structure, which intercepts charged particles, destroys the symmetry of the device, and leads to reduced confinement of plasma. Alternatively, the support structure can be eliminated by use of superconducting rings which are levitated by use of magnetic fields, but requirements to shield the superconductor from the high energy fusion neutrons are formidable.
It is possible to have a separatrix magnetic surface which encloses two or more current channels, of which one is the plasma and the rest are conductors external to the plasma. In this case the separatrix still encloses two or more sets of nested flux surfaces, however there is plasma in only one set of nested flux surfaces. The hyperbolic magnetic axis occurs on the separatrix where there are the more than two possible directions in which the magnetic field line may be followed. The shape of the nested flux surfaces in a toroidal cross section through the surfaces, referred to as a plasma cross section, is normally a set of concentric circles for a reversed field pinch. If a separatrix is formed near the RFP, the hyperbolic magnetic axis occurs on the separatrix where the flux surface crosses itself. The interface between the plasma and the surrounding vacuum is a magnetic flux surface and is referred to as the plasma surface.
Another confinement principle is shown in the copending application of the present inventor, filed Dec. 14, 1981, for Multipole Pinch Method and Apparatus for Producing Average Magnetic Well in Plasma Confinement. The device there shown can be considered as a multipole device in which the solid internal rings have been replaced by high current pinch plasma current channels. Just as in the solid ring multipole devices, approximately equal currents flowing in parallel through the plasma current channels generate a hyperbolic magnetic axis and separatrix magnetic surfaces internal to the plasma. This produces an average magnetic well, provided the component of magnetic field in the direction of the hyperbolic magnetic axis is not too large in the vicinity of the hyperbolic magnetic axis, which can always be achieved by operating the plasma current channels like reversed field pinches so that q=0 occurs in the vicinity of such hyperbolic magnetic axis.